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Reports on Mathematical Physics

Partition Function for 1-Dimensional Kitaev Chain Model Engendering the Majorana Zero Mode

Cheng Da, Hony-Yi Fan

Published Apr 1, 2025• 0 citations
Publisher
DOI
10.1016/s0034-4877(25)00021-7

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Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.
The Kitaev chain has been extensively explored in the context of uniform couplings, with studies focusing either on purely nearest-neighbor interactions or on systems dominated by long-range superconducting pairing. Building on these investigations, we introduce a hybrid Kitaev chain in which the lattice is partitioned into two segments: the left segment comprises nearest-neighbor couplings, while the right segment incorporates long-range pairing. To probe the role of the interface, we study two scenarios: a decoupled (suppressed hopping) case, where the segments are isolated, and a coupled case, where they are connected via interface hopping that enables tunneling. Using this setup, we investigate the behavior of Majorana zero modes at the interface between the two segments, finding that in the decoupled case, Majorana zero modes remain sharply localized at the left segment chain edges while massive Dirac modes remain in right segment chain edges, with their energies and localization strongly dependent on the long-range pairing exponent. Introducing a finite interface coupling enables transfer of Majorana zero modes from the edges of the left segment to those of the right segment of the chain. We characterize this dynamics by the fidelity of state transfer, dynamical rotation, and inverse participation ratio. We show the signature of Majorana zero mode transfer across the interface by the spatiotemporal profile of the probability distribution of the time evolved state.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the "trivial" and "topological" regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes - the nonuniform Ising-Kitaev chain, containing two "topological" regions separated by a "trivial" region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are "delocalized" between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the "nearest-neighbor" form of the effective low-energy Hamiltonian.